The bank doesn't just sit on that money. I mean they might for a while, but eventually they have to loan it out because otherwise they are just acting as a giant, well protected, and completely free (in most cases) wall safe for the customer - which costs them money.

Is it possible that banks would have so much cash from people stashing their money in savings that they might just stop taking deposits? Or would we just see negative interest?

Interesting thing i found:
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At the start of the 21st century, that same personal savings figure dropped from a positive 174.3 billion dollars in 2004, down to a negative 34.8 billion dollars in 2005. The following year in 2006, that figure went down even further when it reached negative 102.8 billion dollars.

General economic principles state that consumers will save some of their disposable income and spend the rest. A negative savings rate means that U.S. consumers are spending more than 100% of their monthly after-tax disposable income . Other variables aside, the overall decline of personal savings (as has been consistently calculated over the past 80 years) indicates that the percentage of household savings has gone down to zero and is now in negative territory.
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Source: US Consumers: Negative Savings, Trillions In Debt, Negative Equity, Disaster Ahead « Chicken Smith

The Obama administration is betting that an extra $8 a week in most Americans' paychecks will boost consumer spending and help pull the U.S. out of its downturn.

One piece of the $787 billion economic recovery package, which President Barack Obama plans to sign Tuesday in Denver, is an experiment in consumer behavior. The $116 billion in tax credits for 95% of Americans will come largely through reduced tax withholding from paychecks, over two years, rather than one-time payments.

The idea: let money trickle out to consumers so it feels like a permanent income boost.

The Obama administration is betting that an extra $8 a week in most Americans' paychecks will boost consumer spending and help pull the U.S. out of its downturn.

One piece of the $787 billion economic recovery package, which President Barack Obama plans to sign Tuesday in Denver, is an experiment in consumer behavior. The $116 billion in tax credits for 95% of Americans will come largely through reduced tax withholding from paychecks, over two years, rather than one-time payments.

The idea: let money trickle out to consumers so it feels like a permanent income boost.

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